mohanty59supriya

## Answered question

2022-07-16

Prove that if x is a rational number and y is an irrational number,then x+y is an irrational number. If in addition, x≠0, then show that xy is an irrational number.

### Answer & Explanation

Jeffrey Jordon

Expert2022-11-07Added 2605 answers

Hint: Consider $\left(x+y\right)-x$

Explanation:

As is very often the case, we do not need to write this as a proof by contradiction. We can prove the contrapositive directly.

We can prove directly:

x is rational $⇒$ (x+y is rational $⇒$ y is rational)

(using $a,b\in Q⇒a-b\in Q$ -- that is, Q is closed under subtraction)

Therefore (by contraposition of the imbedded conditional)

x is rational $⇒$ (y is not rational $⇒$ x+y is not rational)

This is logically equivalent to

(x is rational & y is not rational) $⇒$ x+y is not rational)

By contradiction

Suppose p and $¬q$ and r .

Prove a contradiction and conclude that if p and $¬q$, then $¬r$

By contrapositive

Suppose p and $¬r$. Prove that q.

Conclude that If p and $¬q$, then r.

The two methods are very closely related and I don't know of anyone who accepts one and not the other. (Although many/most/all intuitionists refuse to accept either contradiction or contrapositive.)

Proof by contrapositive

Suppose that x is rational and $x+y$ is rational.

Then the difference $\left(x+y\right)-x=y$ is rational.

Hence is we know that y is irrational, then $x+y$ must have been irrational.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?